Distribution networks are operated in different conditions and the traditional protection systems might not be proper for all available operation conditions by using directional overcurrent relaying. Besides, the presence of distributed generations makes the protection setting and coordination more complex, and may even make it impossible to access a reliable protection system. Therefore, a fully distributed adaptive protection scheme based on the multi-agent systems is proposed to improve the reliability, selectivity and sensitivity of relays. It uses only numerical directional overcurrent relays with instantaneous characteristics as agents, so that only need to pick-up settings and react quickly. All agents dynamically update their pickup value to increase the fault detection sensitivity, and make reliable and selective on-board decisions for fault location and isolation. In addition, in order to achieve high sensitivity and mitigate miscoordination, using setting groups of relays, a novel adaptive backup protection scheme, which does not need complex computational coordination, is proposed. The proposed system is performed in the distributed manner to enhance efficiency and robustness against cyber-attacks and one-point failures. The proposed system is tested on the meshed IEEE 14-bus system and radial IEEE 13-bus system with distributed generations to evaluate its performance.